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Final Report AF Contract 4462067C0049 January 1, 1967 to December 31, 1973 The participants with their current positions are as follows: K.L. Chung, Stanford University John B. Walsh, University of British Columbia Charles Lamb, University of British Columbia Erhan Cinlar, Northwestern University Peter Brockwell, La Trobe University (Australia) The following persons completed their doctorates under the super vision of Chung during the period of the project, listed below with their current positions: Arthur Pittenger, University of Maryland Robert Smythe, University of Washington Gaston Giroux, University of Sherbrooke (Canada) Michael Chamberlain, University of Santa Clara Pending completion of his doctorate is Chris Nevison. Short term consultants include the following persons: H. Kesten, D. Stroock, D. Austin, N. Jain, S. Lloyd, J. L. Doob, C. Dellacherie, D. Burkholder, A. Garsia. Publications of the participants are listed below: K. L. Chung (1) (With John B. Walsh) "To reverse a Markov process," Acta Math., Vol. 123, pp. 225251, 1969. (2) Boundary Theory for Markov Chains, xvi+94 pages, Princeton University Press, 1970. (3) "On diverse questions of time reversing in Markov chains (and processes)," Prod. of the Twelfth Biennial Seminar Canadian Congress of Math., pp. 165175. (4) "Boundary behavior of Markov chains and its contributions to general processes," Invited lecture at the International Congress of Mathe maticians, Nice, 1970; Actes du Congrbs, Vol. 2, pp. 499506, 1971. Final Report Chung 2 (5) "A simple proof of Doob's convergence theorem," Seminaire de Probabilites V, Universite de Strasbourg, SpringerVerlag, 1971. (6) "On the fundamental hypotheses of Hunt processes," Symposia Mathematica, Istituto Nazionale di Alta Matematica, Vol. IX, pp. 4352, 1972. (7) "Poisson process as renewal process," Period. Mat., Vol. 2, pp. 4148, 1972. (8) "An expression for canonical entrance laws," (to appear). (9) "Some universal field equations," S&minaire de Probabilites VI, pp. 9097, Universite de Strasbourg, SpringerVerlag, 1972. (10) "Crudely stationary counting processes", Amer. Math. Monthly, vol. 79, pp. 867877, 1972. (11) "Probabilistic approach to the equilibrium problem in potential theory" (to appear in Ann. Inst. Fourier). (12) (With Brockwell) "Emptiness times for a dam with stable imput and general release function," (to appear). John B. Walsh (1) "The Martin boundary and completion of Markov chains," Z. Wahr scheinlichkeitstheorie und Verw. Gebiete, Vol. 14, pp. 169199, 1970. (2) "Some remarks on the Feller property," Ann. Math. Statist., Vol. 41, pp. 16721683, 1970. (3) "Time reversal and the completion of Markov processes," Invent. Math., Vol. 10, pp. 5781, 1970. Charles W. Lamb (1) "On the construction of certain transition functions," Ann. Math. Statist., Vol. 42, pp. 439450, 1971. (2) "Decomposition and construction of Markov chains," Z. Wahrschein lichkeitstheorie und Verw. Gebiete, Vol. 19, pp. 213224, 1971. (3) "A note on harmonic functions and martingales," Ann. Math. Statist., Vol. 42, pp. 20442049, 1971. Erhan Cinlar (1) "Theory of continuous storage with Markov additive inputs and a general release rule," J. Math. Anal. Appl., Vol. 43, pp. 207231, 1973. (2) (With Jagers) "Two mean values which characterize the Poisson process," J. Appl. Probaility, Vol. 10, pp. 678681, 1973. Final Report Chung 3 Peter Brockwell (1) "On the spectrum of a class of matrices arising in storage theory," Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 25, pp. 253260, 1973. (2) "Deviations from monotonicity of a Wiener process with drift," (to appear in J. Appl. Probability). Detailed descriptions of work done have been included in the annual status reports. To summarize, the major achievements are: (I) Boundary theory for Markov chains, see under Chung (2) and (4); (II) Time reversing of Markov processes, see under Chung (1); and Walsh (2); (III) Application of last exit distribution to equilibrium problem, see under Chung (11). (IV) Various applications to Poisson and point processes, see under Chung (7), (10) and (12). It was further planned to apply the techniques in treating boundaries, time reversals and last exits to practical problems such as turbulence, electromagnetic equilibrium, dam and other storage problems. Initial progress was made in Chung and Brockwell (see under Chung (12)). Termi nation of the project has cut it short. Professor Kai Lai Chung Principal Investigator Unclassified Security Classification DOCUMENT CONTROL DATA R & D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) l2a. REPORT SECURITY CLASSIFICATION Mathematics Department Unclassified Stanford University 2b. GROUP Stanford, CA 94305 3. REPORT TITLE RESEARCH IN PROBABILISTIC TECHNIQUES FOR SYSTEMS ANALYSIS (MARKOV PROCESSES) 4. DESCRIPTIVE NOTES (Type of report and Inclusive dates) Scientific Final S. AUTHOR(S) (First name, middle Initial, last name) Kai Lai Chung S. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1974 3 22 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBERS) F 4462067C0049 b. PROJECT NO. 9769 C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. 10. DISTRIBUTION STATEMENT 1I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Office of Scientific Research Tech, Other 1400 Wilson Blvd. Arlington, VA 22209 13. ABSTRACT Major results are obtained for the boundary theory of Markov chains, time reversing of Markov processes, and application of last exit distribution to equilibrium problems. Practical applications are envisaged for Poisson and point processes; turbulence, electromagnetic and storage problems. al \ae . DD .."" 1441723 Unclassified Security Classlficapfon October 8, 1974 Dr. William G. Rosen Program Director for Modern Analysis and Probability National Science Foundation Washington, D.C. 20550 Dear Dr. Rosen: The following is a summary of work performed to date on NSF grant 41710X, as well as a request for continued funding for 197576. A budget for the period March 1, 1975 to Februairy 28, 1976 is attached. Owing to the circumstance that Dr. Charles Lamb was unable to participate, it is estimated that there will be an unspent balance of approximately $3,000 in the current funds. During the summer of 1974, Dr. John B. Walsh and I worked together for two months. We finished a paper cont dining a new and simpler proof of P. A. Meyer's celebrated theorem that a stopping time is predictable where the sample path is continuous in a Hunt process. As a byproduct we found a single family of stopping times which announce a predictable time simultaneously for all probability laws. Our next area of investi gation is the relation between equilibrium, energy and duality. I found that if the potential equilibrium density is symmetric, then the equi librium measure obtained by the last exit method (Chung, 1972) in fact has the minimizing properties as in the classical cases of Gauss and Frostman. The asymmetric case seems much harder but a similar variational approach has been suggested by discrete analogues in Markov chains. I was invited to give a talk on the subject by the London Mathematical Society at the Durham Symposium on Functional Analysis and Stochastic Processes, where I learned of some new possibilities of approaching .ihe problem. Work will be continued in this direction. Dr. Walsh did very substantial work on duality theory by introducing a new method based on Doob's hpath process but coupled with the idea of last exits near the lifetime. He has completed a paper entitled, "The cofine topology revisited" which will be published. We intend to persue various applications of this method. Dr. Walsh will be in Europe next summer, but can come for short periods. We propose to have Dr. Pierre van Moerbeke participate in the project for two months in the summer of 1975. He will be visiting associate professor in the mathematics department here and fits in very well with the program. He studied with Kac and McKean and will bring his knowledge of diffusion processes and differential equations to bear on the analytic applications of modern probability theory. He will also work on optimal boundary problems. His vitae and list of publications are enclosed herewith for your evaluation and approval. I plan to take a sabbatical leave in the academic year of 197576. Stanford University pays half of my salary for the period. I am Dr. William G. Rosen requesting your support for one sixth of my academic salary which will allow me to have two quarters of leave. The change in my academic salary reflects an annual increase of approximately 5% effective September 1, 1974. A similar increase can be projected for September 1, 1975. During my leave I plan to visit several universities in the east and abroad. I have applied for a Guggenheim fellowship to make this possible. There is no other pending application for other support. If you wish any further information regarding this report and renewal request, please let me know. Thanking you for your support, I am, Sincerely yours, Kai Lai Chung Professor of Mathematics KIIC:ca Atts. October 8, 1974 Three reprints each of ##3, 4, 7 above are hereby enclosed, as well as a copy of #9. Reprints of ##l and 2 were sent to you previously. Reprints ##5 and 11 should be arriving soon and will be sent. Summaries of previous work were given in my annual reports for 1974 and 1975. The work done by Dr. Raq and myself during 1976 was reported in the new proposal submitted to you last October. Dr. Rao obtained a nice proof of Kanda's noted result about polar sets for Levy processes, using the energy principle. My work on a modified energy concept for probabilistic potential theory is continuing. The work on certain Wiener functionals associated with the Schridinger equations, initiated in the onedimensional case in #9 above, is now being extended to the much more difficult case of high dimensions by Dr. Varadhan and myself. The work on Brownian excursions and related topics (##2, 4, 5, 7) is further developing. Lecture notes on them are being prepared and expanded by Dr. Balkema and myself for publication in monograph form. K, L. Chung, Principal Investigator STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS October 14, 1975 Dr. William G. Rosen Program Director for Modern Analysis and Probability National Science Foundation Washington, D.C. 20550 Reference: NSF Grant No. MPS7400405 A01 Dear Dr. Rosen: This is a summary of work performed to date on NSF grant MPS7400405 A01, and a request for continued funding for 19761977. A budget for the period March 1, 1976 February 28, 1977 is attached. The following papers have been published, accepted and submitted for publication: 1. K. L. Chung and John B. Walsh: "Meyer's theorem on predictability", Z. Wahrscheinlichkeitstheorie 29 (1974), 253256. 2. K. L. Chung: "Maxima in Brownian excursions", Bull. Amer. Math. Soc. 81 (1975), 742745. 3. K. L. Chung: "Remarks on equilibrium potential and energy", Ann. Inst. Fourier (in print). 4. K. L. Chung: "A bivariate distribution in regeneration", J. Appl. Probability (in print). 5. K. L. Chung: "Excursions in Brownian motion" (submitted for publication). 6. K. L. Chung and R. Durrett: "Upcrossing and local time" (to appear). 7. John B. Walsh: "The cofine topology revisited" (to appear). 8. Pierre van Moerbeke: "The spectrum of Jacobi matrices" (to appear). 9. Michael Steele: "Combinatorial entropy and uniform limit laws", Doctoral Dissertation, Stanford University, 1975. Reprints and preprints of the works listed above, except 9, are being sent to you under separate cover. Mr. Steele received some support from the project, obtained his doctorate and is now at the University of British Columbia. The following manuscripts are being prepared: 10. K. L. Chung: "On the condenser problem" 11. K. L. Chung and P. van Moerbeke: "Brownian functional related to a classical differential equation". For the summer of 1976, support for Professor Murali Rao and Mr. Richard Durrett is requested. At a recent conference I learned that Rao is working on problems concerning the application of probability methods to potential Dr. W. G. Rosen, NSF, Washington theory, very much along the lines of my project. After seeing my previous results on equilibrium potential, he is eager to participate in further investigations. He is writing a book on Brownian motion and potentials, and his knowledge and experience of the field will be most valuable. Mr. Durrett is a student in the Department of Operations Research at Stanford and has attended several of my courses and seminars. He is the best student in probability that we have had in a number of years. He has already co operated with me on a finished paper (# 6 above) and has started on another (# 10 above). He has essentially completed his doctoral dissertation (under the supervision of Professor Iglehart) and would like to continue the work on my project before taking an academic position. Vitae for Rao and Durrett are herewith submitted for your approval. If you wish any further information regarding this report and renewal request, please let me know. Thanking you for your support, I am, Sincerely yours, Kai Lai Chung Professor of Mathematics KLC:ca Encls. 101475 Report of Progress for Summer 1976 As mentioned in my earlier proposal for Summer 76, the subject of probabilistic potential theory has received renewed interest. This was the reason for initiating a research project on ClassicalcumModern Potential Theory which will culminate in a book that will delineate the classical material which has had such intensive ramifications. The grant has helped me develop a major position of the project. The exploration of related questions has led to extension of known results concerning the maximum principle and the representation of potentials. In another direction Professor Chung and myself have been working on the implications of the energy principle. The ideas we have developed will lead to a generalization and simplification of a recent major result of M. Kanda, namely that for some Levy processes semipolar sets are polar. STANFORD UNIVERSITY STANFORD. CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS October 18, 1977 Dr. William G. Rosen National Science Foundation Washington, D.C. 20550 Dear Dr. Rosen: The following is a summary of the research done by Dr. Murali Rao and myself during the past summer on NSF grant MCS7701319. (1) The notion of energy related to a potential density kernel u(x,y) is extended from the symmetric case to the general case by a modification as follows. For a smooth compact K with its equilibrium measure v, as previously established in [1], the lastexit kernel L(x,dy) = u(x,y)v(dy) considered on K has a stationary probability measure r such that irL = w. Define p(y) = f 7r(dx)u(x,y) > 0 and put l u (x,y) = u(x,y) (y) When u is symmetric, 'p reduces to a constant, which may be taken to be one. In the general case v minimizes the energy I (X) = ffX(dx)u (x,y)X(dy) among all signed measures X of finite variation, which are supported by K and correspond to continuous additive functionals. Further consequences of the modification are being studied in order to put energy concepts for nonsymmetric kernels on a similar footing with the classical theory in the symmetric case. (2) In an earlier paper [2], Rao has given the Riesz representation of excessive functions in the setting of [1]. Under the additional assumption (a) u(x,y) is bounded when x and y vary over disjoint compact sets; he now proves the uniqueness of the representation. From this he deduces that Hunt's Hypothesis (B) is satisfied. Finally, under the further assumption (b) there exists a strictly positive continuous function such that y f m(dx)f(x)u(x,y) belongs to Co; he proves that there is a dual which is a Feller process. Thus we now have a class of Markov processes in duality under easily verifiable conditions. I have simplified some of the proofs and compared the results with those obtained by John B. Walsh in a quite different setting. For instance, in the latter approach u(.,y) is assumed to be minimal excessive for each y, whereas this fact can be deduced from our assumptions. These very recent results will be written up for publication after further consolidation. Dr. W. G. Rosen (3) Jointly with Dr. Varadhan, I have completed some work on diffusion and the Schrbdinger equation. This utilizes the techniques of stochastic integrals and improves upon previous results by Dr. van Moerbeke and myself. A preprint of the paper is enclosed. The extension to high dimensions has already begun with the cooperation of Varadhan. (4) A paper by Rao, entitled "On a result of M. Kanda", which was started in the summer of 1976 on our project, has been accepted for publication in the Zeitschrift fUr Wahrscheinlichkeitstheorie. Thanking you for your support, Sincerely yours, Kai Lai Chung [1] K. L. Chung, Probabilistic approach in potential theory to the equilibrium problem, Ann. Inst. Fourier 23 (1973), 313322. [2] Murali Rao, Excessive functions as potentials of measures, J. London Math. Soc. (to appear). 101877 STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS September 26, 1977 Dr. William G. Rosen National Science Foundation Washington, D.C. 20550 Dear Dr. Rosen: The following is a summary of the research done by Dr. Murali Rao and myself during the past summer on NSF grant MCS7701319. (1) The notion of energy related to a potential density kernel u(x,y) is extended from the symmetric case to the general case by a modification as follows. For a smooth compact K with its equilibrium measure v, as previously established in [1], the lastexit kernel L(x,dy) = u(x,y)v(dy) considered on K has a stationary probability measure w such that 7rL = f. Define p?(y) = f r(dx)u(x,y) > 0 and put 1 u (x,y) = u(x,y)p(y)  When u is symmetric, ( reduces to a constant, which may be taken to be one. In the general case P minimizes the energy I (X) = ffX(dx)up(x,y)X(dy) among all signed measures X of finite variation, which are supported by K and correspond to continuous additive functionals. Further consequences of the modification are being studied in order to put energy concepts for nonsymmetric kernels on a similar footing with the classical theory in the symmetric case. (2) In an earlier paper [2], Rao has given the Riesz representation of excessive functions in the setting of [1]. Under the additional assumption (a) u(x,y) is bounded when x and y vary over disjoint compact sets; he now proves the uniqueness of the representation. From this he deduces that Hunt's Hypothesis (B) is satisfied. Finally, under the further assumption (b) there exists a strictly positive continuous function such that y > f m(dx)f(x)u(x,y) belongs to Co; he proves that there is a dual which is a Feller process. Thus we now have a class of Markov processes in duality under easily verifiable conditions. I have simplified some of the proofs and compared the results with those obtained by John B. Walsh in a quite different setting. For instance, in the latter approach u(.,y) is assumed to be minimal excessive for each y, whereas this fact can be deduced from our assumptions. These very recent results will be written up for publication after further consolidation. Dr. W. G. Rosen (3) Jointly with Dr. Varadhan, I have completed some work on diffusion and the Schridinger equation. This utilizes the techniques of stochastic integrals and improves upon previous results by Dr. van Moerbeke and myself. A preprint of the paper is enclosed. The extension to high dimensions has already begun with the cooperation of Varadhan. (4) A paper by Rao, entitled "On a result of M. Kaida", which was started in the summer of 1976 on our project, has been accepted for publication in the Zeitschrift fUr Wahrscheinlichkeitstheorie. Thanking you for your support, Sincerely yours, Kai Lai Chung [1] K. L. Chung, Probabilistic approach in potential theory to the equilibrium problem, Ann. Inst. Fourier 23 (1973), 313322. [2] Murali Rao, Excessive functions as potentials of measures, J. London Math. Soc. (to appear). 92677 Final Report This is the final report on NSF grant MCS 7400405A02, March 1, 1976 to February 28, 1977. The participants are: K. L. Chung, Stanford University John B. Walsh, now at University of British Columbia Pierre van Moerbeke, now at University of Orsay Murali Rao, University of Aarhus The grant also supported in part the following doctorate students: Chris Nevison, now at Colgate University Mike Steele, now at University of British Columbia Richard Durrett, now at University of California in Los Angeles The following research papers were produced: 1. K. L. Chung and John B. Walsh: "Meyer's theorem on predictability", Z. Wahrscheinlichkeitstheorie 29 (1974), 253256. 2. K. L. Chung: "Maxima in Brownian excursions", Bull. Amer. Math. Soc. 81 (1975), 742745. 3. K. L. Chung: "Remarks on equilibrium potential and energy", Ann. Inst. Fourier 25 (1975), 131138. 4. K. L. Chung: "A bivariate distribution in regeneration", J. Appl. Probability 12 (1975), 837839. 5. K. L. Chung: "Excursions in Brownian motion", Arkiv fUr Matematik 14 (1976), 155177. 6. K. L. Chung and R. K. Getoor: "The condenser problem", to appear in Ann. of Probability. 7. K. L. Chung and R. Durrett: "Downcrossings and local time", Z. Wahr scheinlichkeitstheorie 35 (1976), 147150. 8. K. L. Chung: "A proof of Skorohod's lemma", Teor. Verojatnost. i Mat. Statist. 15 (1976), 151152. 9. K. L. Chung and Pierre van Moerbeke: "On certain Wiener functionals associated with the Schrddinger equation", submitted for publication. 10. John B. Walsh: "The cofine topology revisited", to appear. 11. Pierre van Moerbeke: "The spectrum of Jacobi matrices", Invent. Math. 37 (1976), 4581. 12. Murali Rao: "On a result of M. Kanda", to appear. Three reprints each of ##3, 4, 7 above are hereby enclosed, as well as a copy of #9. Reprints of ##1 and 2 were sent to you previously. Reprints ##5 and 11 should be arriving soon and will be sent. Summaries of previous work were given in my annual reports for 1974 and 1975. The work done by Dr. Rao and myself during 1976 was reported in the new proposal submitted to you last October. Dr. Rao obtained a nice proof of Kanda's noted result about polar sets for Levy processes, using the energy principle. My work on a modified energy concept for probabilistic potential theory is continuing. The work on certain Wiener functionals associated with the Schridinger equations, initiated in the onedimensional case in #9 above, is now being extended to the much more difficult case of high dimensions by Dr. Varadhan and myself. The work on Brownian excursions and related topics (##2, 4, 5, 7) is further developing. Lecture notes on them are being prepared and expanded by Dr. Balkema and myself for publication in monograph form. K, L. Chung, Principal Investigator STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 DEPARTMENT OF MATHEMATICS September 11, 1978. Dr. William G. Rosen Program Director for Modern Analysis and Probability National Science Foundation Washington, D.C. 20550 Reference: NSF Grant No. MCS7701319 A01 Dear Dr. Rosen: The following is a summary of the research work done during the past few months on NSF grant MCS701319 A01. (1) Dr. K. Murali Rao and I made a thorough investigation of the class of potential kernels introduced in [1]. We deal with a transient Hunt process with a potential density function u satisfying the two conditions below: (a) y + u(x,y)1 is finite continuous for each x; (b) u(x,y) = if and only if x = y. We proved a number of results under these conditions. Some of these were known previously only under strong duality and strong Feller assumptions. Our approach has the advantage of tying simple verifiable analytic conditions to deep proba bilistic conclusions, thereby revealing structural relations sometimes "snowed under" by other postulates. A basic result is the general existence of a version w of the density u having the property that for each open set G: w(x,y) = PGw(x,y) for all y in G where PG is the balayage operator acting on x. Under (a) and (b) we showed that there is a polar set Z such that w(.,y) = u(.,y) or = 0 according as y Z or y E Z. This enables us to prove the uniqueness of the measure p determining the potential Up, provided that p does not charge Z. A Riesz decomposition is an easy consequence. Furthermore, we prove that Hunt's Hypothesis (B) holds in our setting, namely: for any open G containing any compact K we have P K1 = PK1 ; provided that u(,y) is excessive for each y. This turns out to be a difficult proposition. We are now encouraged to attack another one of Hunt's Hypothesis: "Every semipolar set is polar", which goes back to Kellogg, and is a cornerstone of the classical theory. (2) Along the way we established the existence of a dual semigroup from which a right continuous strong Markov dual process can be constructed. It is strong Feller under a time change and satisfies the continuity principle. Yet it does not have all the properties under the usual duality assumptions. We intend to investigate the relation of this dual process to the samplepath reverse which always exists according to the result by Chung and Walsh (1969), and also the utransform studied by Doob and Walsh (1976). Dr. W. G. Rosen, NSF, Washington (3) As a consequence of Hypothesis (B) mentioned above, it follows that if v is the equilibrium measure on the compact K obtained by the method of [1], the set function K * v(K) induces a true Choquet capacity. Pre viously I proved this under the additional assumption that all paths are continuous. Further development in this direction, in particular a theory of energy as initiated in [4], depends on a better understanding of the modified potential kernel introduced there. The results summarized above partially supplant earlier ones reported last October. Two preprints [2] and [3] have been sent to you in June and a third one [4] is being sent under separate cover. These are being combined and rewritten in a paper entitled "A new setting for potential theory", which is under preparation. This will be presented by me in an invited talk in a special session at the Claremont AMS meeting in October. Mr. Joseph Glover was supported by the grant during the spring quarter before he obtained his doctorate from UCSD La Jolla. (He is now lecturer in the Department of Statistics at UC Berkeley.) Paper [5] is a joint effort by him and me which was sent to you in June and has been submitted for publication. In it a number of wellknown results for the "right" process are established for the "left" process. More sophisticated methods are needed because several standard arguments are no longer valid in the left case, but it came to us as a pleasant surprise that the famous theorems by CartanBrelotDoob, Hunt, Meyer and Dellacherie all hold true without any sacrifices. As Meyer said, it certainly helps to be able to use the left as well as the right hand. One good application has already been pointed out by Rao, more should be forthcoming. Thanking you for your support, Sincerely, Kai Lai Chung Professor of Mathematics KLC:ca [1] K. L. Chung, Probabilistic approach in potential theory to the equilibrium problem, Ann. Inst. Fourier 23 (1973), 313322. [2] K. L. Chung and Murali Rao, Potential theory without duality, Aarhus University Preprint Series No. 22 (19778). [3] K. L. Chung and Murali Rao, On existence of a dual process, ibid. No. 25 (19778) [4] K. L. Chung and Murali Rao, Equilibrium and energy (to be sent). [5] K. L. Chung and Joseph Glover, Left continuous moderate Markov processes (submitted for publication). 91178 Progress Report MCS8301072 The following papers have been completed during the past year. [1] K.L. Chung, The gauge and conditional gauge theorem. Seminaire de probabilites XIX (1983/84), 496503. [2] K.L. Chung, DoublyFeller process with multiplicative functional, Seminar on Stochastic Processes, vol. 5, Birkhauser (to appear). [3] M. Liao, Riesz representation and duality of Markov processes, Doctoral Dissertation, Stanford University, August 1984. In [1] it is proved that the conditional gauge theorem, which is an important strengthening of the gauge theorem, actually follows from the latter provided an estimate concerning a cordon sanitaire of small measure holds true. (See under item (1) in the new research proposal.) Recently Chung gave a talk on this at the Institute of Mathematics and its Applications at the University of Minnesota, pin pointing the problem. Afterwards, Cranston, Fabes, and Zhao were able to establish the required estimate for a bounded Lipschitz domain. This great improvement over previous results (first for C2, then for C11 domains) was made possible by the reduction contained in [1]. In [2] it is proved that if a process has both the Feller and strong Feller properties, then, if we attach a certain class of multiplicative functional to the process and kill it off a regular open (not necessarily bounded!) set, the resulting process has the same properties. In the special case of Brownian motion, this result forms the base of analytic developments of the FeynmanKac formula. The new result does not require the continuity of paths and covers a wide class of multiplicative functionals. In [3] Liao generalized and simplified a number of previous results by Chung and Rao. In particular, assumptions on the potential density function u(x,y) are considerably reduced, and the existence of the exceptional set Z is thoroughly 2 investigated. In the duality case this is identified as the set of branching points. Liao's thesis has many points of contact with current research in Europe on axiomatic potential theory. He has returned to China and is now on the faculty of Nankai University, Tianjin, after teaching for one semester at the University of Florida. Copies of [1] and [2] are being sent to your office. Reprint of [3] will be sent when available. Progress Report MCS8301072 During the month of July, 1986 Chung worked with Thomas Salis bury on the extension of the gauge theorem to an arbitrary domain inEuclidean space with Martin boundary. It turned out that the methods used by Chung in his paper "The Gauge and Conditional Gauge Theorem" (Sgminaire des probabilit4s vol. XIX, 1983/84) works equally well in the general case with appropriate modifica t. ions supplied by the construction of Martin boundary. See also obeJ the separate progress report submitted by Salisbury. ,, i Chung and Z. Zhao also proved the continuity of theAgauge function up to the boundary. This and a number of related results will be published in a monograph under preparation, tentatively entitled "From Brownian Motion to Schrbdinger equation". The following note deals with one of the problems originally proposed in the project: K. L. Chung, Remark on the conditional gauge theorem A preprint of this and a reprint of the following are sent to your office under separate cover. K. L. Chung, P. Lis and R. J. Williams, Comparison of probability and classical methods for the Schridinger equation, Expo. Math vol. 4 (1986), 271278.  This is a substantially revised version of an older manuscript. Yr It deals with the eigenvalue (spectobn) connection of the gauge b'theorem, miae a problem in the original proposal. Although hatxx results Tor a larger class of functions q have been obtained (to be publghed in the pe.o ,ee monograph mentioned above), the ana lytic connection is essentially the same. A doctoral student, V. Papanicolaou is working on the mixed boundary problem under the direction of Chung. 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